A new symmetry criterion based on the distance function and applications to PDE's

We prove that, if Omega subset of R-n is an open bounded starshaped domain of class C-2, the constancy over partial derivative Omega of the function phi(y) = integral(lambda(y))(0) Pi(n-1)(j=1) [1 - t kappa(j)(y)]dt implies that Omega is a ball. Here kappa(j)(y) and lambda(y) denote respectively the principal curvatures and the cut value of a boundary point y is an element of partial derivative Omega. We apply this geometric result to different symmetry questions for PDE's: an overdetermined system of Monge-Kantorovich type equations (which can be viewed as the limit as p -> +infinity of Serrin's symmetry problem for the p-Laplacian), and equations in divergence form whose solutions depend only on the distance from the boundary in some subset of their domain. (c) 2013 Elsevier Inc. All rights reserved.